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February  2010, 27(1): 205-216. doi: 10.3934/dcds.2010.27.205

$C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings

Shaobo Gan 1, , Kazuhiro Sakai 2, and  Lan Wen 3,

1. 

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871
2. 

Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505
3. 

School of Mathematic Sciences, Peking University, Beijing, 100871

Received  June 2009 Revised  November 2009 Published  February 2010

Let $f$ be a diffeomorphism of a closed $n$-dimensional $C^\infty$ manifold, and $p$ be a hyperbolic saddle periodic point of $f$. In this paper, we introduce the notion of $C^1$-stably weakly shadowing for a closed $f$-invariant set, and prove that for the homoclinic class $H_f(p)$ of $p$, if $f_{|H_f(p)}$ is $C^1$-stably weakly shadowing, then $H_f(p)$ admits a dominated splitting. Especially, on a 3-dimensional manifold, the splitting on $H_f(p)$ is partially hyperbolic, and if in addition, $f$ is far from homoclinic tangency, then $H_f(p)$ is strongly partially hyperbolic.
Keywords: dominated splitting, partially hyperbolic., shadowing, Weak shadowing, pseudo-orbit, homoclinic class, chain component, chain recurrent set.
Mathematics Subject Classification: Primary: 37B20, 37C29, 37C50, 37D20, 37D3.
Citation: Shaobo Gan, Kazuhiro Sakai, Lan Wen. $C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 205-216. doi: 10.3934/dcds.2010.27.205
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